Unless I'm wrong, K_0 is in general neither left nor right exact, and so it doesn't make sense to talk about its derived functors. But at least in topological K-theory, and the K-theory of C*-algebras, the K_0 and K_1 groups fit into an exact hexagon.
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Saul GlasmanJan 11 '10 at 13:59

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If you consider K_0 as functor Schemes -> Ab Groups, what do you mean by derived functors? (Schemes is not an abelian category)
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Evgeny ShinderJan 11 '10 at 19:40

3 Answers
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I don't think it's stupid, but I guess it depends what you mean by "derived functor." This is true in the weak sense that K-theory is naturally a space- or spectrum-valued functor, and the $K_i$ is the i-th homotopy of this functor. But it seems not to be the case that $K$-theory is a derived functor in the sense of Cartan-Eilenberg.

Let me discuss the question of the universality of $K$-theory:

I'll abuse terminology and refer to "categories" when I mean categories of a suitable kind, with appropriate added structure --- e.g., exact categories if you want to do Quillen K-theory, Waldhausen categories if you want to do Waldhausen K-theory, Waldhausen $\infty$-categories if you want to do K-theory with them, etc. ...

Now if one translates the sense in which $K_0$ is universal as an abelian-group-valued functor on "categories" into the language of stable homotopy theory, one arrives at the universal property satisfied by K-theory as a spectrum-valued functor on "categories."

More precisely, we have additive $K_0$, denoted $K_0^{\oplus}$, which is simply the functor that assigns to any "category" $\mathcal{C}$ the group completion of the abelian monoid whose elements are isomorphism (or equivalence) classes of objects of $\mathcal{C}$, where the sum is $\oplus$. This functor is "inadequate" in the sense that there might be some exact (or fiber) sequences of $\mathcal{C}$ that $K_0^{\oplus}$ cannot see.

To address this, for any "category" $\mathcal{C}$, we can build a new "category" $\mathcal{E}(\mathcal{C})$ whose objects are exact sequences. This "category" admits two functors to $\mathcal{C}$ that send an exact sequence $[0\to A'\to A\to A''\to 0]$ to either $A'$ or $A''$. For any functor $F$ from categories to abelian groups, we get an induced homomorphism $F\mathcal{E}(\mathcal{C})\to F\mathcal{C}\oplus F\mathcal{C}$. Let's say that $F$ splits the exact sequences of $\mathcal{C}$ if this morphism is an isomorphism, and let's say that $F$ is additive if $F$ splits the exact sequences of every "category."

Now $K_0$ has the following pleasant universal property. It is the initial object in the category of additive functors receiving a natural transformation from $K_0^{\oplus}$.

Now to translate all this into stable homotopy. We have additive K-theory, denoted $K^{\oplus}$, which is simply the functor that assigns to any "category" $\mathcal{C}$ the spectrum corresponding to the group completion of the $E_{\infty}$ space given by the (nerve of the) subcategory of $\mathcal{C}$ comprised of the isomorphisms (or weak equivalences), where the sum is $\oplus$. This functor is again "inadequate" in the sense that there might be some exact (or fiber) sequences of $\mathcal{C}$ that $K^{\oplus}$ cannot see.

Now for any functor $F$ from categories to spectra, we get an induced homomorphism $F\mathcal{E}(\mathcal{C})\to F\mathcal{C}\vee F\mathcal{C}$. Let's say that $F$ splits the exact sequences of $\mathcal{C}$ if this morphism is an equivalence, and let's say that $F$ is additive if $F$ splits the exact sequences of every "category."

Now $K$ has the following homotopy-universal property. It is the homotopy-initial object in the category of additive functors receiving a natural transformation from $K^{\oplus}$.

So the universality of K-theory arises not from thinking of the disembodied K-groups, but rather from interpreting K-theory as a spectrum, and rewriting the universal property of $K_0$ in suitably homotopical language.

(References: Gonçalo Tabuada has a paper in which he characterizes K-theory by a similar universal property, and John Rognes and I have begun a similar paper in the context of Waldhausen $\infty$-categories, an incomplete draft of which is on my webpage.)

Could you post that link to your work with John Rognes here? Or else send me something by email. I can't see it on your website. In fact, I can't see anything on your website! :-) Except a text at the very bottom that tells me to click on "any link below" of which I can see none. Sorry, maybe I am missing something...
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Urs SchreiberFeb 18 '10 at 19:41

There is a definition due Sasha Rosenberg, of a variant of the algebraic K-theory which is universal practically by definition and in the setting much more general than Quillen exact categories.
By universal I mean something along Cartan-Eilenberg and Tohoku.
He first defines the notion of a right exact structure on a category, what is just a Grothendieck pretopology whose covers are singletons which are strict epimorphisms. Now the collection of all small categories with right exact structure has a left exact structure, what is a dual notion (Grothendieck precotopology which...). Now he extends the formalism of Tohoku to define universal delta functors or delta star functors (I never know which one is which) for right or left exact structures. So, one can define K-zero by hand, on the collection of all categories with right exact structures and try to extend it to a universal delta functor. And there is such. In particular, Quillen exact categories have a canonical right exact structure. Now this variant of K-theory has all other standard properties of Quillen K-theory, like resolution by devissage, exactness and so on. But it is not clear if it is equal or not to Quillen K-theory yet (for this one should evaluate Quillen recipe on the injective resolution by the categories with right exact structures; if one gets zero, voila!). The article of Rosenberg is at

The algebraic $K$-groups of a commutative unital ring can indeed be defined as derived functors, but one needs to work in the context of non-abelian homological algebra in the sense of A. Dold, D. Puppe, Homologie nicht-additiver Funktoren, Ann. Inst. Fourier 11 (1961), and
M. Tierney, W. Vogel: Simplicial resolutions and derived functors, Math. Zeit. 111 (1969).
A useful introduction is given in the book `Non-abelian homological algebra and its applications' by Hvedri Inassaridze (Kluwer, 1997).

For a group $G$, define $\displaystyle Z_\infty (G)=\lim_{\leftarrow} G/\Gamma_i(G)$,
where $\{\Gamma_i(G)\}$ is the lower central series of $G$. This defines a functor $Z_\infty: Gr\rightarrow Gr$.
Now Theorem 5.1 in the cited book roughly reads as follows:

Let $L_*Z_\infty$ be the left derived functors of the functor $Z_\infty$. Then $L_i Z_\infty(GL(R))$ is isomorphic to Quillen's $K_i(R)$.